324 ANALYTICAL MECHANICS
249. Logarithmic Decrement. — The logarii hm of the ratio of two consecutive amplitudes is constant and is called the logarithmic decrement of the motion. The amplitudes occur whenever the relation
tan (at) = — a
is satisfied. Let the first amplitude occur at the instant t = h; then since the period of the tangent is TT, the times of the succeeding amplitudes are given by
tan (wf) = tan (wti + HT) , u / / _L 7l7r
or by t = Cl + ~~ »
where n is a positive integer. Hence, denoting the logarithmic decrement by X and the nth amplitude by anj we have
X = log —2- (by definition)
a» + 2
- ^i - Sin^wfi ±
" " "'""
A6""flvl " ...... « v sin [wfe + (n + 2)
! 6 V w/
= log - r-TTTT
= ~P. (XXV)
Therefore if I is known k" may be determined from observations of P and a.
* Obtained by setting^- » 0.